50 research outputs found
Optimal distributed control of a generalized fractional Cahn-Hilliard system
In the recent paper `Well-posedness and regularity for a generalized
fractional Cahn-Hilliard system' (arXiv:1804.11290) by the same authors,
general well-posedness results have been established for a a class of
evolutionary systems of two equations having the structure of a viscous
Cahn-Hilliard system, in which nonlinearities of double-well type occur. The
operators appearing in the system equations are fractional versions in the
spectral sense of general linear operators A,B having compact resolvents, which
are densely defined, unbounded, selfadjoint, and monotone in a Hilbert space of
functions defined in a smooth domain. In this work we complement the results
given in arXiv:1804.11290 by studying a distributed control problem for this
evolutionary system. The main difficulty in the analysis is to establish a
rigorous Frechet differentiability result for the associated control-to-state
mapping. This seems only to be possible if the state stays bounded, which, in
turn, makes it necessary to postulate an additional global boundedness
assumption. One typical situation, in which this assumption is satisfied,
arises when B is the negative Laplacian with zero Dirichlet boundary conditions
and the nonlinearity is smooth with polynomial growth of at most order four.
Also a case with logarithmic nonlinearity can be handled. Under the global
boundedness assumption, we establish existence and first-order necessary
optimality conditions for the optimal control problem in terms of a variational
inequality and the associated adjoint state system.Comment: Key words: fractional operators, Cahn-Hilliard systems, optimal
control, necessary optimality condition
Deep quench approximation and optimal control of general Cahn-Hilliard systems with fractional operators and double obstacle potentials
The paper arXiv:1804.11290 contains well-posedness and regularity results for
a system of evolutionary operator equations having the structure of a
Cahn-Hilliard system. The operators appearing in the system equations were
fractional versions in the spectral sense of general linear operators A and B
having compact resolvents and are densely defined, unbounded, selfadjoint, and
monotone in a Hilbert space of functions defined in a smooth domain. The
associated double-well potentials driving the phase separation process modeled
by the Cahn-Hilliard system could be of a very general type that includes
standard physically meaningful cases such as polynomial, logarithmic, and
double obstacle nonlinearities. In the subsequent paper arXiv:1807.03218, an
analysis of distributed optimal control problems was performed for such
evolutionary systems, where only the differentiable case of certain polynomial
and logarithmic double-well potentials could be admitted. Results concerning
existence of optimizers and first-order necessary optimality conditions were
derived. In the present paper, we complement these results by studying a
distributed control problem for such evolutionary systems in the case of
nondifferentiable nonlinearities of double obstacle type. For such
nonlinearities, it is well known that the standard constraint qualifications
cannot be applied to construct appropriate Lagrange multipliers. To overcome
this difficulty, we follow here the so-called "deep quench" method. We first
give a general convergence analysis of the deep quench approximation that
includes an error estimate and then demonstrate that its use leads in the
double obstacle case to appropriate first-order necessary optimality conditions
in terms of a variational inequality and the associated adjoint state system.Comment: Key words: Fractional operators, Cahn-Hilliard systems, optimal
control, double obstacles, necessary optimality condition
Energy consistent discontinuous Galerkin methods for a quasi-incompressible diffuse two phase flow model
We design consistent discontinuous Galerkin finite element schemes for the approximation of a quasi-incompressible two phase flow model of Allen--Cahn/Cahn--Hilliard/Navier--Stokes--Korteweg type which allows for phase transitions. We show that the scheme is mass conservative and monotonically energy dissipative. In this case the dissipation is isolated to discrete equivalents of those effects already causing dissipation on the continuous level, that is, there is no artificial numerical dissipation added into the scheme. In this sense the methods are consistent with the energy dissipation of the continuous PDE system
Optimal distributed control of a diffuse interface model of tumor growth
In this paper, a distributed optimal control problem is studied for a diffuse
interface model of tumor growth which was proposed in [A. Hawkins-Daruud, K.G.
van der Zee, J.T. Oden, Numerical simulation of a thermodynamically consistent
four-species tumor growth model, Int. J. Numer. Math. Biomed. Engng. 28 (2011),
3-24]. The model consists of a Cahn-Hilliard equation for the tumor cell
fraction coupled to a reaction-diffusion equation for a variable representing
the nutrient-rich extracellular water volume fraction. The distributed control
monitors as a right-hand side the reaction-diffusion equation and can be
interpreted as a nutrient supply or a medication, while the cost function,
which is of standard tracking type, is meant to keep the tumor cell fraction
under control during the evolution. We show that the control-to-state operator
is Frechet differentiable between appropriate Banach spaces and derive the
first-order necessary optimality conditions in terms of a variational
inequality involving the adjoint state variables.Comment: A revised version of the paper has been published on Nonlinearity 30
(2017), 2518-2546. Let us point out that in this arXiv:1601.04567 [math.AP]
version there is something missing in assumption (H3) at page 6: the first
initial value in (H6) must also satisfy a Neumann homogeneous condition at
the boundary of the domai
Distributed optimal control of a nonstandard nonlocal phase field system
We investigate a distributed optimal control problem for a nonlocal phase
field model of viscous Cahn-Hilliard type. The model constitutes a nonlocal
version of a model for two-species phase segregation on an atomic lattice under
the presence of diffusion that has been studied in a series of papers by P.
Podio-Guidugli and the present authors. The model consists of a highly
nonlinear parabolic equation coupled to an ordinary differential equation. The
latter equation contains both nonlocal and singular terms that render the
analysis difficult. Standard arguments of optimal control theory do not apply
directly, although the control constraints and the cost functional are of
standard type. We show that the problem admits a solution, and we derive the
first-order necessary conditions of optimality.Comment: 38 Pages. Key words: distributed optimal control, nonlinear phase
field systems, nonlocal operators, first-order necessary optimality
condition
Uniqueness of viscosity solutions of Local Cahn-Hilliard-Navier-Stokes system
In this work, we consider the local Cahn-Hilliard-Navier-Stokes equation with
regular potential in two dimensional bounded domain. We formulate distributed
optimal control problem as the minimization of a suitable cost functional
subject to the controlled local Cahn-Hilliard-Navier- Stokes system and define
the associated value function. We prove the Dynamic Programming Principle
satisfied by the value function. Due to the lack of smoothness properties for
the value function, we use the method of viscosity solutions to obtain the
corresponding solution of the infinite dimensional Hamilton-Jacobi-Bellman
equation. We show that the value function is the unique viscosity solution of
the Hamilton-Jacobi-Bellman equation. The uniqueness of the viscosity solution
is established via comparison principle